Irreducible representationIn mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper nontrivial subrepresentation , with closed under the action of . Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.
Lie algebra representationIn the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.
Weight (representation theory)In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space.
Representation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).
Semisimple representationIn mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations). It is an example of the general mathematical notion of semisimplicity. Many representations that appear in applications of representation theory are semisimple or can be approximated by semisimple representations.
Algebra representationIn abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra.
Modular representation theoryModular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field K of positive characteristic p, necessarily a prime number. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.
BrandA brand is a name, term, design, symbol or any other feature that distinguishes one seller's good or service from those of other sellers. Brands are used in business, marketing, and advertising for recognition and, importantly, to create and store value as brand equity for the object identified, to the benefit of the brand's customers, its owners and shareholders. Brand names are sometimes distinguished from generic or store brands.
Apple Inc.Apple Inc. is an American multinational technology company headquartered in Cupertino, California. Apple is the world's largest technology company by revenue, with in 2022 revenue. , Apple is the world's biggest company by market capitalization. As of June 2022, Apple is the fourth-largest personal computer vendor by unit sales and the second-largest mobile phone manufacturer in the world. It is considered one of the Big Five American information technology companies, alongside Alphabet (parent company of Google), Amazon, Meta Platforms, and Microsoft.
Place brandingPlace branding (includes place marketing and place promotion) is a term based on the idea that "cities and regions can be branded," whereby branding techniques and other marketing strategies are applied to "the economic, political and cultural development of cities, regions and countries." As opposed to the branding of products and services, place branding is more multidimensional in nature, as a 'place' is inherently "anchored into a history, a culture, an ecosystem," which is then incorporated into a network of associations, "linking products, spaces, organizations and people.
